Theorem spw 2035

Description: Weak version of the specialization scheme sp 2186. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2186 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2186 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2138 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2186 are spfw 2034 (minimal distinct variable requirements), spnfw 1980 (when 𝑥 is not free in ¬ 𝜑), spvw 1982 (when 𝑥 does not appear in 𝜑), sptruw 1807 (when 𝜑 is true), spfalw 1981 (when 𝜑 is false), and spvv 1989 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1911	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1911	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1911	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2034	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781

This theorem is referenced by:  hba1w  2050  19.8aw  2053  exexw  2054  spaev  2055  ax12w  2136  rspw  3209  reldisj  4400  ralidmw  4455  dtruALT2  5306  bj-ssblem1  36698  bj-ax12w  36721  eu6w  42779